# 0.10 Lab 7b - discrete-time random processes (part 2)

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Questions or comments concerning this laboratory should be directedto Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907;(765) 494-0340; bouman@ecn.purdue.edu

## Bivariate distributions

In this section, we will study the concept of a bivariate distribution.We will see that bivariate distributions characterize how two random variables are related to each other.We will also see that correlation and covariance are two simple measures of the dependencies between random variables,which can be very useful for analyzing both random variables and random processes.

## Background on bivariate distributions

Sometimes we need to account for not just one random variable, but several. In this section, we will examine the case of two randomvariables–the so called bivariate case–but the theory is easily generalized to accommodate more than two.

The random variables $X$ and $Y$ have cumulative distribution functions (CDFs) ${F}_{X}\left(x\right)$ and ${F}_{Y}\left(y\right)$ , also known as marginal CDFs. Since there may be an interaction between $X$ and $Y$ , the marginal statistics may not fully describe their behavior.Therefore we define a bivariate , or joint CDF as

${F}_{X,Y}\left(x,y\right)=P\left(X\le x,Y\le y\right).$

If the joint CDF is sufficiently “smooth”, we can define a joint probability density function,

${f}_{X,Y}\left(x,y\right)=\frac{{\partial }^{2}}{\partial x\partial y}{F}_{X,Y}\left(x,y\right).$

Conversely, the joint probability density function may be used to calculate thejoint CDF:

${F}_{X,Y}\left(x,y\right)={\int }_{-\infty }^{y}{\int }_{-\infty }^{x}{f}_{X,Y}\left(s,t\right)ds\phantom{\rule{0.166667em}{0ex}}dt.$

The random variables $X$ and $Y$ are said to be independent if and only if their joint CDF (or PDF) is a separable function, which means

${f}_{X,Y}\left(x,y\right)={f}_{X}\left(x\right){f}_{Y}\left(y\right)$

Informally, independence between random variables means that one random variable does not tell you anything about the other.As a consequence of the definition, if $X$ and $Y$ are independent, then the product of their expectations is the expectation of their product.

$E\left[XY\right]=E\left[X\right]E\left[Y\right]$

While the joint distribution contains all the information about $X$ and $Y$ , it can be very complex and is often difficult to calculate. In many applications, a simple measure of the dependenciesof $X$ and $Y$ can be very useful. Three such measures are the correlation , covariance , and the correlation coefficient .

• Correlation
$E\left[XY\right]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }xy{f}_{X,Y}\left(x,y\right)dx\phantom{\rule{0.166667em}{0ex}}dy$
• Covariance
$E\left[\left(X-{\mu }_{X}\right)\left(Y-{\mu }_{Y}\right)\right]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(x-{\mu }_{X}\right)\left(y-{\mu }_{Y}\right){f}_{X,Y}\left(x,y\right)dxdy$
• Correlation coefficient
${\rho }_{XY}=\frac{E\left[\left(X-{\mu }_{X}\right)\left(Y-{\mu }_{Y}\right)\right]}{{\sigma }_{X}{\sigma }_{Y}}=\frac{E\left[XY\right]-{\mu }_{X}{\mu }_{Y}}{{\sigma }_{X}{\sigma }_{Y}}$

If the correlation coefficient is 0, then $X$ and $Y$ are said to be uncorrelated . Notice that independence implies uncorrelatedness,however the converse is not true.

## Samples of two random variables

In the following experiment, we will examine the relationship between the scatter plots for pairs of random samples $\left({X}_{i},{Z}_{i}\right)$ and their correlation coefficient. We will see that the correlation coefficient determines the shape ofthe scatter plot.

Let $X$ and $Y$ be independent Gaussian random variables, each with mean 0 and variance 1. We will consider the correlation between $X$ and $Z$ , where $Z$ is equal to the following:

1. $Z=Y$
2. $Z=\left(X+Y\right)/2$
3. $Z=\left(4*X+Y\right)/5$
4. $Z=\left(99*X+Y\right)/100$

Notice that since $Z$ is a linear combination of two Gaussian random variables, $Z$ will also be Gaussian.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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